I must say that SmokeDetector seems to be a really useful thing.

These two questions are quite similar: $X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$ and Show that $ \limsup x_n ≤ \limsup y_n$ and $\liminf x_n ≤ \liminf y_n$.

The first one specifically says that the asker uses definition of limsup based on limit of subsequences.

In the other one, the OP does not specify the definition. The answer given there uses $\limsup_{n\to\infty} x_n = \lim\limits_{n\to\infty} \sup\limits_{k\ge n} x_k$. (An answer using such definition is also used in an answer to the first question.)

What is the best solution? Merge? Close one of them as a duplicate?

I think it's safe to close the second one as a duplicate of the first

Are there some more duplicates which I did not find?

Ok then

Let's go with this direction. Voted.

I also updated the first question to make clear what the link between $E$ and the lim sup/inf was

That seems reasonable. (I should have noticed it when I was editing the title.)

dup'ed

Here is another duplicate: math.stackexchange.com/questions/1063786/…

also dup'ed

Another one?

And another one? they're both proof verification requests, so I don't know what to do.

You're good at searching for questions.

I found everything through the "Related" sidebar :)

Oh, I should have thought about that.

Discussions on meta: Proof-verification and duplicates and Closing as a duplicate if the post contains OP's own proof/solution.

Funny identites sounds like the epitome of "Too broad/primarily opinion based."

Someone mentioned earlier that a good way to deal with such things is to have a moderator lock as "of historical interest". Is there any interest in doing so, or should I vote?

@NajibIdrissi I think that in both cases the OP has received some feedback on their attempt. So I will vote to close as a duplicate.

That makes sense @Martin

@MikeMiller Vote. As elaborated here (When is it appropriate ...), historical locks should only be applied to questions having been closed/reopened or deleted/undeleted at least once.

Thanks. Done.

close This is a duplicate -- I commented but I'm out of close votes.

2 votes now.

dup'ed

And deleted by OP. Whatever, it wasn't adding value.